Gradient Fields

Modern NMR spectrometers are equipped with the capability of applying a magnetic field gradient in any direction for a chosen, brief amount of time. If the direction is along the sample's $z$-axis, then while the gradient is on, the field varies as $B(z)=B_0+\gamma z B_1$, where $B_0$ is the strong, external field and $B_1$ is the gradient power. As a result of this gradient, the precession frequency of nuclear spins depends on their positions' $z$-coordinates. One of the most important applications of gradients is NMR imaging because gradients make it possible to distinguish different parts of the sample.

The effect of applying a $z$-gradient can be visualized for the situation in which there is only one observable nuclear spin per molecule. Suppose that the initial deviation density matrix of each nuclear spin is $\sigma_x$ in the rotating frame. After a gradient pulse of duration $t$, the deviation of a nuclear spin at position $z$ is given by $e^{-i\sigma_z\nu z t/2}\sigma_x e^{i\sigma_z\nu z t/2} = \cos(\nu zt)\sigma_x+\sin(\nu zt)\sigma_y$, where the constant $\nu$ depends linearly on the strength of the gradient and the magnetic moment of the nucleus. See Fig. 8. The effect of the gradient is a $z$-dependent change in phase. The coil used to measure planar magnetization integrates the contribution to the magnetization of all the nuclei in the neighborhood of the coil. Assuming a coil equally sensitive over the interval between $-a$ and $a$ along the sample's $z$-axis, the observed total $x$-magnetization is:

$$M_x = \int_{-a}^{a}dz\,\mbox{tr}\left(\sigma_x(\cos(\nu zt)\sigma_x+\sin(\nu zt)\sigma_y)\right)$$

$$= \int_{-a}^{a}dz\,\mbox{tr}\left(\cos(\nu zt)\sigma_x^2+\sin(\nu zt)\sigma_x\sigma_y\right)$$

$$= \int_{-a}^{a}dz\,\mbox{tr}\left(\cos(\nu zt)+i\sin(\nu zt)\sigma_z\right)$$

$$= 2\int_{-a}^a dz\cos(\nu zt).$$ (18)

For large values of $\nu t$, $M_x\simeq 0$. In general, a sufficiently powerful gradient pulse eliminates the planar magnetization.

$\stackrel{\textrm{\large gradient}}{\longrightarrow}$

FIG. 8: Effect of a pulsed gradient field along the $z$ axis in the rotating frame. Initial $x$-magnetization is assumed. A spin at $z=0$ is not affected, but the ones above and below are rotated by an amount proportional to $z$. As a result, the local planar magnetization follows a spiral curve.

Interestingly, the effect of a gradient pulse can be reversed if an opposite gradient pulse is applied for the same amount of time. This effect is called a ``gradient echo''. The reversal only works if the second pulse is applied sufficiently soon. Otherwise, diffusion randomizes the molecules' positions along the gradient's direction before the second pulse. If the positions are randomized, then the phase change from the second pulse is no longer correlated with that from the first for any given molecule. The loss of memory of the phase change from a gradient pulse can be fine-tuned by variations in the delay between the two pulses in a gradient echo sequence. This method can be used for applying a controllable amount of phase noise, which is useful for investigating the effects of noise and the ability to correct for noise in QIP.

If the gradient pulse is not reversed and the memory of the phase changes is lost, then the pulse's effect can be described as an irreversible operation on the state of the nuclear spin. If the initial state of the nuclear spin in each molecule is $\rho$, then after the gradient pulse, the spin state of a molecule at position $z$ is given by $\rho(z)= e^{-i\sigma_z\nu z t/2}\rho e^{i\sigma_z\nu z t/2}$. Suppose that the positions of the molecules are randomized over the region that the coil is sensitive to. Now it is no longer possible to tell where a given molecule was when the gradient pulse was applied. As a result, as far as our observations are concerned, the state of a molecule is given by $\rho(z)$, where $z$ is random. In other words, the state is indistinguishable from

$$\rho'={1\over 2a}\int_{-a}^{a}dz \rho(z) ={1\over 2a}\int_{-a}^{a}dz e^{-i\sigma_z\nu z t/2}\rho e^{i\sigma_z\nu z t/2}.$$ (19)

Thus the effect of the gradient pulse is equivalent to the operation $\rho\rightarrow\rho'$ as defined by the above equation. This is an operation of the type mentioned at the end of the previous section and can be used for making states such as pseudopure states. Note that after the gradients have been turned off, nuclei at different positions cannot be distinguished by the measurement coil. It is therefore not necessary to wait for the molecules' positions to be randomized.

So far we have described the effects of gradient pulses on isolated nuclear spins in a molecule. In order to restrict the effect to a single nuclear spin in a molecule, one can invert the other spins between a pair of identical gradient pulses in the same direction. This technique refocuses the gradient for the inverted spins. An example of how effects involving multiple nuclear spins can be exploited is the algorithm for pseudopure state preparation described in Sect. 3.2.